Math 2.0

Number theory Divisibility and primes Theorem

Euclid–Euler theorem

Builds on Perfect numberMersenne primeSum of divisors

Statement

Let nn be an even positive integer. Then nn is a perfect number if and only if n=2p−1(2p−1)n = 2^{p-1} (2^p - 1), where 2p−12^p - 1 is a Mersenne prime.

Remarks

The primes 2p−12^p - 1 for p=2,3,5,7p = 2, 3, 5, 7 give the perfect numbers 66, 2828, 496496 and 81288128. So the even perfect numbers correspond one to one to the Mersenne primes. The proof uses the multiplicativity of the sum of divisors: σ(2p−1q)=(2p−1)(q+1)\sigma(2^{p-1} q) = (2^p - 1)(q + 1) for an odd prime qq.

History

Euclid proved in the Elements that 2p−1(2p−1)2^{p-1}(2^p - 1) is perfect when 2p−12^p - 1 is prime. Ibn al-Haytham conjectured the converse around 1000 CE. Euler proved the converse in the 18th century, in a paper published after his death.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

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Arithmetic function, definition, no evidence levelArithmetic functiondefinitionDivisibility, definition, no evidence levelDivisibilitydefinitionMersenne number, definition, no evidence levelMersenne numberdefinitionDivisors of a natural number, definition, no evidence levelDivisors of a natural…definitionPrime number, definition, no evidence levelPrime numberdefinitionDivisor function, definition, no evidence levelDivisor functiondefinitionMersenne prime, definition, no evidence levelMersenne primedefinitionSum of divisors, definition, no evidence levelSum of divisorsdefinitionPerfect number, definition, no evidence levelPerfect numberdefinitionEuclid–Euler theorem, statement, no evidence levelEuclid–Euler theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Arithmetic function definition
  2. Divisibility definition
  3. Mersenne number definition
  4. Divisors of a natural number definition
  5. Prime number definition
  6. Divisor function definition
  7. Mersenne prime definition
  8. Sum of divisors definition
  9. Perfect number definition
  10. Euclid–Euler theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Euclid%E2%80%93Euler_theorem, Statement and examples
  2. Web https://en.wikipedia.org/wiki/Euclid%E2%80%93Euler_theorem, History
About this page
Tag0WETJCKNRG
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Kindstatement, theorem
WritingW1 stated
MSC11A25
StatusThis copy of the site has no build data. Run m2 status 0WETJCKNRG.
UsesPerfect number, Mersenne prime, Sum of divisors
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